algebra | Notion

群groups、环rings、域fields、向量空间vector spaces、模modules (over a ring)、域上的代数algebra over a field、格lattices（偏序集partially ordered set）⇒ 代数结构

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algebraic structures: sets with specific operations acting on their elements.

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代数结构(as objects)及其同态homomorphisms (as arrows) ⇒ 数学范畴 mathematical category

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同态homorphisms a map between two algebraic structures of the same type, that preserver the operations of the structures.

$$ f(\mu_A(x,\cdots,y))=\mu_B(f(x),\cdots, f(y))\\ f:A\to B $$

a map $f:A\to B$ preserves an operation $\mu$ of arity $k$, defined on both $A$ and $B$

映射保留操作$\mu$，此映射为同态(映射)。

$\operatorname{Hom}(A,B)$表示所有同态构成的集合

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同构isomorphism

双射同态。（一一对应，满射）

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酉空间与线性空间同构
- 同胚
  
  可以看作拓扑学上的同构
- 自同构 automorphism
  - $\operatorname{Aut} G$ 群的自同构全体，与（函数）复合运算构成群$G$的自同构群
    
    automorphism group symmetry group of the object
    
    https://en.wikipedia.org/wiki/Automorphism

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category a collection of “objects” that are linked by “arrows“

to study properties and constructions that are similar for various structure

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原群：封闭性

a non-empty set $G$ together with a binary operation on $G$.封闭性$a\cdot b\in G$

【Group axioms】

a set $R$ with 2 binary operations, addition $(x,y)\mapsto x+y$, and multiplication $(x,y)\mapsto xy$,

含幺环(乘法恒元，单位半群)⇒可逆环(乘法逆元，群)⇒域(乘法交换律，阿贝尔群) 交换环(乘法交换律) 【数域】

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